Quivers of groups with normal p-subgroups

Let K be a sufficiently large field of characteristic p. We determine the quiver of a semidirect product H = P ⋊ H̄ of a finite p-group by a finite group H̄, as the join of the quiver of H̄ and the McKay graph D(H̄, R̄′) of the conjugation representation R̄′ of H̄ on Rad(KP)/(Rad KP ∩ Rad² (KH)). More generally, whenever P ⊴ H and H̄ →^∼ H/P, we show that the quiver Q_KH of H is a subgraph of the above join, and give a necessary and sufficient condition on the radicals for the quiver of H to exactly equal the join Q_KH̄ v D(H̄, R̄′). Finally, we identify the "transgressing" arrows of the McKay graph, those which do not appear in the quiver of H, with basis elements in the kernel of the transgression map.